The second comparison was proved in [1]} in a multidimensional setting. The one-dimensional case was rediscovered, e.g., in [2]}. Optimal constants are known for some arguments (in the lower bound for \(F_p\)  with \(p\in (1,2)\)  and the upper bound for \(p\in (2,\infty )\) ), see [3]} and [4]}.
The first comparison in (REF ) follows from the second one. A historical discussion of the third comparison in (REF ) is given [5]}.
Many special cases of (REF ) can be found in the earlier works [6]}, [7]}, [8]}; we refer to [9]} for a full proof.
